Question

Identify the following partial differential equations in the given order based on their generic form where \( c \) is a constant, and \( u_1 \) and \( u_2 \) are functions of \( (x, t) \) and \( (x, y) \), respectively:
(i) \[ \frac{\partial^2 u_1}{\partial t^2} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (ii) \[ \frac{\partial u_1}{\partial t} = c^2 \frac{\partial^2 u_1}{\partial x^2} \] (iii) \[ \frac{\partial^2 u_2}{\partial x^2} + \frac{\partial^2 u_2}{\partial y^2} = 0 \]

• A. (i) Wave equation, (ii) Heat equation, (iii) Laplace equation
• B. (i) Heat equation, (ii) Laplace equation, (iii) Wave equation
• C. (i) Laplace equation, (ii) Wave equation, (iii) Heat equation
• D. (i) Heat equation, (ii) Wave equation, (iii) Laplace equation

Hint:

Match each equation to its standard form: second time derivative for wave, first time derivative for heat, and pure spatial Laplacian equal to zero for Laplace.

Answer 1

The Correct Option is A

Step 1: Recall the three standard PDE forms.
Wave equation: $u_{tt} = c^2 u_{xx}$, which has two time derivatives.
Heat equation: $u_t = c^2 u_{xx}$, which has one time derivative.
Laplace equation: $u_{xx} + u_{yy} = 0$, which has no time variable and two space variables.

Step 2: Compare equation (i).
$u_{1tt} = c^2 u_{1xx}$ has a second-order time derivative on the left, so it matches the wave equation form exactly.

Step 3: Compare equation (ii).
$u_{1t} = c^2 u_{1xx}$ has only a first-order time derivative, which is the mark of the heat equation.

Step 4: Compare equation (iii).
$u_{2xx} + u_{2yy} = 0$ has two independent space variables and no time term at all, so it is the Laplace equation.

Final Answer:
The order (i), (ii), (iii) is wave, heat, Laplace. \[ \boxed{\text{(A)}} \]